Calculus: Special Edition: Chapters 5-8, 11, 12, 14

Front Cover
Title
Copyright
Contents
Preface
For the Student
For the Instructor
Features of this Book
Text Content
Innovative Presentation
Acknowledgments
5 Integration
5.1 Antidifferentiation
Reversing Differentiation
Antiderivative Notation
Antidifferentiation Formulas
Applications
Area as an Antiderivative
Problem Set 5.1
5.2 Area as the Limit of a Sum
Area as the Limit of a Sum
The General Approximation Scheme
Summation Notation
Area Using Summation Formulas
Problem Set 5.2
5.3 Riemann Sums and the Definite Integral
Riemann Sums
The Definite Integral
Area as an Integral
Properties of the Definite Integral
Distance as an Integral
Problem Set 5.3
5.4 The Fundamental Theorems of Calculus
The First Fundamental Theorem of Calculus
The Second Fundamental Theorem of Calculus
Problem Set 5.4
5.5 Integration by Substitution
Substitution with Indefinite Integration
Substitution with Definite Integration
Problem Set 5.5
5.6 Introduction to Differential Equations
Introduction and Terminology
Direction Fields
Separable Differential Equations
Modeling Exponential Growth and Decay
Orthogonal Trajectories
Modeling Fluid Flow Through an Orifice
Modeling the Motion of a Projectile: Escape Velocity
Problem Set 5.6
5.7 The Mean Value Theorem for Integrals; Average Value
Mean Value Theorem for Integrals
Modeling Average Value of a Function
Problem Set 5.7
5.8 Numerical Integration: The Trapezoidal Rule and Simpson's Rule
Approximation by Rectangles
Trapezoidal Rule
Simpson's Rule
Error Estimation
Summary of Numerical Integration Techniques
Problem Set 5.8
5.9 An Alternative Approach: The Logarithm as an Integral
Natural Logarithm as an Integral
Geometric Interpretation
The Natural Exponential Function
Problem Set 5.9
Chapter 5 Review
Chapter 5 Group Research Project
6 Additional Applications of the Integral
6.1 Area Between Two Curves
Area Between Curves
Area Using Vertical Strips
Area Using Horizontal Strips
Problem Set 6.1
6.2 Volume
Method of Cross Sections
Method of Disks and Washers
The Method of Cylindrical Shells
Problem Set 6.2
6.3 Polar Forms and Area
The Polar Coordinate System
Polar Graphs
Summary of Polar-Form Curves
Intersection of Polar-Form Curves
Polar Area
Problem Set 6.3
6.4 Arc Length and Surface Area
The Arc Length of a Curve
The Area of a Surface of Revolution
Polar Arc Length and Surface Area
Problem Set 6.4
6.5 Physical Applications: Work, Liquid Force, and Centroids
Work
Modeling Fluid Pressure and Force
Modeling the Centroid of a Plane Region
Volume Theorem of Pappus
Problem Set 6.5
6.6 Applications to Business, Economics, and Life Sciences
Future and Present Value of a Flow of Income
Cumulative Change: Net Earnings
Consumer's and Producer's Surplus
Survival and Renewal
The Flow of Blood Through an Artery
Problem Set 6.6
Chapter 6 Review
Book Report: To Infinity and Beyond, A Cultural History of the Infinite, by Eli Maor
Chapter 6 Group Research Project
7 Methods of Integration
7.1 Review of Substitution and Integration by Table
Review of Substitution
Using Tables of Integrals
Problem Set 7.1
7.2 Integration By Parts
Integration by Parts Formula
Repeated Use of Integration by Parts
Definite Integration by Parts
Problem Set 7.2
7.3 Trigonometric Methods
Powers of Sine and Cosine
Powers of Secant and Tangent
Trigonometric Substitution
Quadratic-Form Integrals
Problem Set 7.3
7.4 Method of Partial Fractions
Partial Fraction Decomposition
Integrating Rational Functions
Rational Functions of Sine and Cosine
Problem Set 7.4
7.5 Summary of Integration Techniques
Problem Set 7.5
7.6 First-Order Differential Equations
First-Order Linear Differential Equations
Applications of First-Order Equations
Problem Set 7.6
7.7 Improper Integrals
Improper Integrals with Infinite Limits of Integration
Improper Integrals with Unbounded Integrands
Comparison Test for Convergence or Divergence
Problem Set 7.7
7.8 Hyperbolic and Inverse Hyperbolic Functions
Hyperbolic Functions
Derivatives and Integrals Involving Hyperbolic Functions
Inverse Hyperbolic Functions
Problem Set 7.8
Chapter 7 Review
Chapter 7 Group Research Project
8 Infinite Series
8.1 Sequences and Their Limits
Sequences
Limits of Sequences
Bounded, Monotonic Sequences
Problem Set 8.1
8.2 Introduction to Infinite Series; Geometric Series
Definition of Infinite Series
General Properties of Infinite Series
Geometric Series
Applications of Geometric Series
Problem Set 8.2
8.3 The Integral Test; p-series
Divergence Test
Series of Nonnegative Numbers; The Integral Test
p-series
Problem Set 8.3
8.4 Comparison Tests
Direct Comparison Test
Limit Comparison Tests
Problem Set 8.4
8.5 The Ratio Test and the Root Test
Ratio Test
Root Test
Problem Set 8.5
8.6 Alternating Series; Absolute and Conditional Convergence
Alternating Series Test
Error Estimates for Alternating Series
Absolute and Conditional Convergence
Summary of Convergence Tests
Rearrangement of Terms in an Absolutely Convergent Series
Problem Set 8.6
8.7 Power Series
Convergence of a Power Series
Term-by-Term Differentiation and Integration of Power Series
Problem Set 8.7
8.8 Taylor and Maclaurin Series
Taylor and Maclaurin Polynomials
Taylor's Theorem
Taylor and Maclaurin Series
Operations with Taylor and Maclaurin Series
Problem Set 8.8
Chapter 8 Review
Chapter 8 Group Research Project
Cumulative Review Problems - Chapters 6-8
11 Partial Differentiation
11.1 Functions of Several Variables
Basic Concepts
Level Curves and Surfaces
Graphs of Functions of Two Variables
Problem Set 11.1
11.2 Limits and Continuity
Open and Closed Sets in R^2 and R^3
Limit of a Function of Two Variables
Continuity
Limits and Continuity for Functions of Three Variables
Problem Set 11.2
11.3 Partial Derivatives
Partial Differentiation
Partial Derivative as a Slope
Partial Derivative as a Rate
Higher-Order Partial Derivatives
Problem Set 11.3
11.4 Tangent Planes, Approximations, and Differentiability
Tangent Planes
Incremental Approximations
The Total Differential
Differentiability
Problem Set 11.4
11.5 Chain Rules
Chain Rule for One Parameter
Extensions of the Chain Rule
Problem Set 11.5
11.6 Directional Derivatives and the Gradient
The Directional Derivative
The Gradient
Maximal Property of the Gradient
Functions of Three Variables
Normal Property of the Gradient
Tangent Planes and Normal Lines
Problem Set 11.6
11.7 Extrema of Functions of Two Variables
Relative Extrema
Second Partials Test
Absolute Extrema of Continuous Functions
Least Squares Approximation of Data
Problem Set 11.7
11.8 Lagrange Multipliers
Method of Lagrange Multipliers
Constrained Optimization Problems
Lagrange Multipliers with Two Parameters
A Geometrical Interpretation of Lagrange's Theorem
Problem Set 11.8
Chapter 11 Review
Book Report: Hypatia's Heritage by Margaret Alic
Chapter 11 Group Research Project
12 Multiple Integration
12.1 Double Integration over Rectangular Regions
Definition of the Double Integral
Properties of Double Integrals
Volume Interpretation
Iterated Integration
An Informal Argument for Fubini's Theorem
Problem Set 12.1
12.2 Double Integration over Nonrectangular Regions
Double Integrals over Type I and Type II Regions
More on Area and Volume
Choosing the Order of Integration in a Double Integral
Problem Set 12.2
12.3 Double Integrals in Polar Coordinates
Change of Variables to Polar Form
Area and Volume in Polar Form
Problem Set 12.3
12.4 Surface Area
Definition of Surface Area
Area of a Surface Defined Parametrically
Problem Set 12.4
12.5 Triple Integrals
Definition of the Triple Integral
Iterated Integration
Volume by Triple Integrals
Problem Set 12.5
12.6 Mass, Moments, and Probability Density Functions
Mass and Center of Mass
Moments of Inertia
Joint Probability Density Functions
Problem Set 12.6
12.7 Cylindrical and Spherical Coordinates
Cylindrical Coordinates
Integration with Cylindrical Coordinates
Spherical Coordinates
Integration with Spherical Coordinates
Problem Set 12.7
12.8 Jacobians: Change of Variables
Change of Variables in a Double Integral
Change of Variables in a Triple Integral
Problem Set 12.8
Chapter 12 Review
Chapter 12 Group Research Project
14 Introduction to Differential Equations
14.1 First-Order Differential Equations
Review of Separable Differential Equations
Homogeneous Differential Equations
Review of First-Order Linear Differential Equations
Exact Differential Equations
Euler's Method
Problem Set 14.1
14.2 Second-Order Homogeneous Linear Differential Equations
Linear Independence
Solutions of the Equation ay" + by' + cy = 0
Higher-Order Homogeneous Linear Differential Equations
Undamped Versus Damped Motion of a Mass on a Spring
Reduction of Order
Problem Set 14.2
14.3 Second-Order Nonhomogeneous Linear Differential Equations
Nonhomogeneous Equations
Method of Undetermined Coefficients
Variation of Parameters
An Application to RLC Circuits
Problem Set 14.3
Chapter 14 Review
Book Report: Mathematical Experience by Philip J. Davis and Reuben Hersh
Chapter 14 Group Research Project
Appendices
A: Introduction to the Theory of Limits
The Believer/Doubter Format
Selected Theorems with Formal Proofs
B: Selected Proofs
Limit Comparison Test (Section 8.4)
Taylor's Theorem (Section 8.8)
Sufficient Condition for Differentiability (Section 11.4)
Change of Variables Formula for Multiple Integration (Section 12.8)
C: Significant Digits
Significant Digits
Rounding and Rules of Computations Used in this Book
Calculator Experiments
Trigonometric Evaluations
Graphing Blunders
D: Short Table of Integrals
E: Trigonometry
Trigonometric Functions
Radians and Degrees
Inverse Trigonometric Functions
Evaluating Trigonometric Functions
Trigonometric Graphs
Trigonometric Identities
Problem Set E
F: Parabolas
Conic Sections
Standard-Form Parabolas With Vertex (0, 0)
Standard-Form Equations of Parabolas
Problem Set F
G: Ellipses
Definition of an Ellipse
Standard-form Ellipse with Center (0, 0)
Standard-form Equations of Ellipses
Eccentricity
Problem Set G
H: Hyperbolas
Definition of a Hyperbola
Standard-form Hyperbola with Center (0, 0)
Standard-form Equations of Hyperbolas
Properties of Hyperbolas
Conic Section Summary
Problem Set H
I: Determinants
Determinants
Properties of Determinants
Problem Set I
J: Answers to Selected Problems
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 11
Chapter 12
Chapter 14
Appendix Answers
Index
Miscellaneous Formulas
Integration Formulas
Differentiation Formulas
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